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Math242E206

# Math242E206 - x = 2 cos t y =-1 sin t for 0 ≤ t ≤ 3...

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FORM B MATH 242, Fall 2006 Exam 2: November 1, 10:10-11:00 Only the blue book will be graded. On the front cover, please write Form B clearly, along with your name and section number. Please start each problem on a new page, circle final answers, and cross out incorrect work. Unless otherwise noted, you must justify all answers to receive full credit. You may not use calculators, notes, or any other kinds of aids. 1. (10 points) Which of the following equations corresponds to the graph at the right? Ex- plain your answer to receive credit. (a) y 2 + 4 y - 4 x 2 = 0 (b) y 2 + 4 y - 4 x = 0 (c) y 2 - 4 y - 4 x 2 = 0 (d) y 2 + 4 y + 4 x 2 = 0 2. (10 points) Convert the polar curve r cos θ = r 2 + 2 r sin θ to rectangular coordinates. 3. (10 points) Find the area inside one loop of the polar curve r = sin 3 θ , from θ = 0 to θ = π / 3. 4. Consider the curve defined by
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Unformatted text preview: x = 2 cos t , y =-1 + sin t , for 0 ≤ t ≤ 3 / 2. (a) (10 points) At what point(s) ( x and y values) does the curve have a vertical tangent? (b) (10 points) Eliminate the parameter and sketch the curve. 5. (15 points each) Evaluate each integral. (You don’t have to combine numerical fractions in the answer.) (a) Z xe 2 x dx (b) Z 1 √ 3 / 2 15 x 3 p 1-x 2 dx 6. (10 points each) Evaluate each integral, or show that it is divergent. (a) Z 3 1 1 √ x-1 dx (b) Z ∞ 1 ln x x dx d dx ‡ sin-1 x · = 1 √ 1-x 2 d dx ‡ tan-1 x · = 1 1 + x 2 1-sin 2 = cos 2 1 + tan 2 = sec 2 sin 2 = 1 2 ( 1-cos 2 ) cos 2 = 1 2 ( 1 + cos 2 )...
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